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XXIII. On the Operating Symbol of Differential Covariants. By 

 The Honourable Chief Justice Cockle, President of the Philo- 

 sophical Society of Queensland*. 



CONSIDERING the complete form of differential cubic, the 

 operating symbol A, of my paper (( On Differential Cova- 

 riants" in the last March (1864) Number of this Journal, will 

 become 



. d d d 



do dc df 



or rather we ought to say 



in order to mark the indissoluble nature of the bond which con- 

 nects the elements, or constituents, of the constituents or ele- - 

 ments of the operator. Moreover a, b, c, /, and x being treated 

 as independent, we have 



or, more fully, 



A — — — A 

 dx~~ dx ' 



AK'=A^- = -^-.AK. 



dx ax 



Illustration will probably serve all the purposes of a more 

 laboured exposition, and the use of the brackets in giving clear- 

 ness and precision to the operations will appear from the follow- 

 ing transformation : 



L dc J ^ ' L del L dcldx~~ dxL del dx" 



If AK vanishes, the commutative property of -y- and A enables 



us to perceive that AK' vanishes also. That AK/ 2 vanishes may 

 be shown directly, thus : 



AK' 2 = &(2bb l - dc- da + ab"-ba") 

 - _ 2ab r - 2bd + 2a! b + 2ab'- aa" + aa" 

 = 0. 



If we employ differentials instead of differential coefficients, 

 the operator for 



{a, b } c,fjd, dxfy 



* Communicated by the Author. 



